Finitness Principles for Smooth Selection

TL;DR

This paper establishes finiteness principles for smooth and Lipschitz selection problems in Euclidean spaces, confirming a longstanding conjecture and advancing the theoretical understanding of selection theorems.

Contribution

It proves finiteness principles for $C^{m}$ and $C^{m-1,1}$-selection, including a proof of Brudnyi-Shvartsman's conjecture for Lipschitz selections on $\,\mathbb{R}^n$.

Findings

Proved finiteness principles for $C^{m}$-selection.

Established finiteness principles for $C^{m-1,1}$-selection.

Provided a proof for Brudnyi-Shvartsman's conjecture on Lipschitz selections.

Abstract

In this paper we prove finiteness principles for $C^{m}\left( \mathbb{R}^{n}, \mathbb{R}^{D}\right) $-selection, and for $C^{m-1,1}\left( \mathbb{R}^{n}, \mathbb{R}^{D}\right) $-selection, in particular providing a proof for a conjecture of Brudyni-Shvartsman (1994) on Lipschitz selections for the case when the domain is $X = \mathbb{R}^n$.